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IWSSIP 2010 - 17th International Conference on Systems, Signals and Image Processing
53
Novel Design of Equiripple Half-Band FIR
Filters
Pavel Zahradnik, Miroslav Vlcek, Boris Simak
Dept. of Communication Engineering, Dept. of Transportation Sciences
Czech Technical University in Prague
Prague, Czech Republic
[email protected], [email protected], [email protected]
Abstract—A novel design procedure of equiripple half-
band FIR filters is developed. Solution of the
approximation problem in terms of generating function
and zero phase transfer function for the equiripple half-
band FIR filter is presented. The equiripple half-band
FIR filters are optimal in the Chebyshev sense. The
generating function for the equiripple half-band FIR
filter is presented. The closed form solution provides an
efficient computation of the impulse response of the
filter. One example is included.
Keywords-FIR filter, half-band filter, equiripple
approximation, optimal filter.
I. INTRODUCTION
Half-band (HB) filters are fundamental building blocks in multirate signal processing. They are used e.g. in filter banks and in compression techniques. The only available method for designing of equiripple (ER) HB FIR filters is based on the numerical McClellan - Parks program [1]. It is usually combined with a clever "Trick" [2]. Besides this, some design methods are available for almost ER HB FIR filters, e.g. [3], [4]. No general non-numerical design of an ER HB FIR filter was found in the references. In our paper we are primarily concerned with the ER approximation of HB FIR filters and with the related non-numerical design procedure suitable for practical design of ER HB FIR filters. We present the generating function and the zero phase transfer function of the ER HB FIR filter. These functions give an insight into the nature of this approximation problem. Based on the differential equation for the Chebyshev polynomials of the second kind, we have derived formulas for an effective evaluation of the coefficients of the impulse response. We present an approximating degree equation that is useful in practical filter design. The advantage of the proposed approach over the numerical design procedures consists in the fact that the coefficients of the impulse response are evaluated by formulas. Hence the speed of the design is deterministic.
II. IMPULSE RESPONSE, TRANSFER FUNCTION AND
ZERO PHASE TRANSFER FUNCTION
An HB filter is specified by the minimal passband
frequency Tpω (or maximal stopband frequency Tsω )
and by the minimal attenuation in the stopband sa [dB]
(or maximal attenuation in the passband pa [dB]). The
antisymmetric behavior of its frequency response implies
the relations TT ps ωπω −= and 1101005.005.0
=+ sp aa . The
goal in the filter design is to get the minimum filter
length N satisfying the filter specification and to
evaluate the coefficients of the impulse response of the filter. We assume the impulse response )(kh with odd
length 1)12(2 ++= nN coefficients and with even
symmetry )1()( kNhkh −−= . The impulse response of
the HB FIR filter contains n2 zero coefficients as
follows
....0),12())12(12(2
...1,0)2()212(2
5.0)0()12(
nkkaknh
nkkaknh
anh
=+=+±+
===±+
==+
(1)
The transfer function of the HB FIR filter is
++= ∑
=
+
+−n
k
k
n TkazzH0
12
)12( )()12(2
1)( ω (2)
where )(wTk is Chebyshev polynomials of the first kind.
The frequency response )( TjeH
ω of the HB FIR filter is
)(cos)( )12( TQeeH TnjTj ωωω +−= (3)
where )(wQ is a polynomial in the variable
2/)( 1−+= zzw which on the unit circle reduces to a
real valued zero phase transfer function )(wQ of the real
argument )cos( Tw ω= .
III. GENERATING POLYNOMIAL AND ZERO PHASE
TRANSFER FUNCTION OF AN ER HB FIR FILTER
The generating polynomial of an ER HB FIR filter is
related to the generating polynomial of the almost ER
HB FIR filter presented in [4]. Based on our
experiments conducted in [4], we have found that the
generating polynomial )(wG (Fig.1) of the ER HB FIR
filter is obtained by weighting of Chebyshev
polynomials in the generating polynomial of the almost
ER HB FIR filter, namely
IWSSIP 2010 - 17th International Conference on Systems, Signals and Image Processing
54
Figure 1. Generating polynomial )(wG for n=20,
03922835.0=′κ , A=1.08532371 and B=0.95360863
Figure 2. Zero phase transfer function )(wQ for n=20,
03922835.0=′κ , A=1.08532371, B=0.95360863 and 55091994.0~
=Ν
′−
′−−+
′−
′−−= − 2
22
12
22
1
12
1
12)(
κ
κ
κ
κ wBU
wAUwG nn
(4)
where )(xUnand )(1 xU n−
are Chebyshev polynomials of
the second kind and A, B, κ ′ are real numbers. The
zero phase transfer function )(wQ (Fig. 2) is related to
the generating polynomial
∫Ν+= dwwGwQ )(~
1
2
1)( (5)
where the norming factor Ν~ is given by (17). Both the
generating polynomial )(wG and the zero phase transfer
function )(wQ show the nature of the approximation of
an ER HB FIR filter.
IV. DIFFERENTIAL EQUATION AND IMPULSE
RESPONSE OF AN ER HB FIR FILTER
The Chebyshev polynomial of the second kind
)(wU xfulfils the differential equation
.0)()2()(
3)(
)1(2
22 =++−− xUnn
dx
xdUx
dx
xUdx n
nn (6)
Using substitution
′−
′−−=
2
22
1
12
κ
κwx (7)
we get the differential equation (6) in the form
[ ]
.0)()2(4
)()1(2)1(
)(3
)()1()(
3
2222
2
2222
=++
−+−′+
−−′−
wUnnw
dw
wdUwww
dw
wdUw
dw
wUdwww
n
n
nn
κ
κ
(8)
Based on the differential equation (8), we have derived the non-numerical procedure for the evaluation of the
impulse response )(khk corresponding to polynomial
corresponding to polynomial )(~
wU n
.1
12)(
~2
22
dww
UwU nn
′−
′−−= ∫ κ
κ (9)
This procedure is summarized in Tab.1. The impulse
response )(kh of the ER HB FIR filter is
).(~)(~2
1)( 1 kh
Bkh
Akh nn −
Ν+
Ν+= (10)
The non-numerical evaluation of the impulse response
)(kh is essential in the practical filter design.
V. DEGREE OF AN ER HB FIR FILTER
The exact degree formula is not available. In the
practical filter design, the degree n can be obtained
with excellent accuracy from the specified minimal
passband frequency Tpω and from the minimal
attenuation in the stopband sa [dB] using the
approximating degree formula
.13196871.2954155181.18
64775300.3318840664.18][
−
+−≥
T
TdBan
p
ps
ω
ω (11)
Figure 3. Emperical dependence of sa [dB] on πω /Tpand n
IWSSIP 2010 - 17th International Conference on Systems, Signals and Image Processing
55
TABLE I. ALGORITHM FOR THE EVALUATION OF THE COEFFICIENTS )(khn
The exact relation between the minimal attenuation in
the stopband sa [dB], the minimal passband frequency
Tpω and the degree n were obtained experimentally.
It is shown in Fig.3. Equation (11) was obtained by the
approximation of exact experimental values in Fig.3.
VI. SECONDARY VALUES OF THE ER HB FIR FILTER
The secondary real values κ ′ , A and B can be
obtained from the specified passband frequency Tpω
and from the degree n of the generating polynomial. In
practical filter design, the approximating formulas
37221484.001927560.1
00665857.057111377.1
+−
+−=′
n
nTn pωκ (12)
n
nnA
73512298.001701407.1
24760314.9036823440.001525753.0
++
′
++= κ
(13)
n
nnB
72759508.002999650.1
75145813.535418408.100233667.0
−+
′
+−= κ
(14)
obtained experimentally provide excellent accuracy.
Further, the exact values κ ′ , A and B can be obtained
numerically (e.g. using the Matlab function fminsearch)
by satisfying the equality (see Fig.4)
=.)(
)1(
)(
01 evenisnifwQ
oddisnifQ
wQ p (15)
The value
12cos)1(
222
01+
′−+′=n
wπ
κκ (16)
was introduced in [4]. Relation (15) guarantees the
equiriple behaviour of the generating polynomial )(wQ .
VII. DESIGN OF THE ER HB FIR FILTER
The design procedure is as follows:
• Specify the ER HB FIR filter by the minimal
passband frequency Tpω and by the minimal
attenuation in the stopband sa [dB].
• Calculate the integer degree n of the generating
polynomial (11).
• Calculate the real values κ ′ (12), A (13) and B (14).
• Evaluate the partial impulse responses )(khnand
)(1 khn− (Tab.1).
• Evaluate the final impulse response )(kh (10)
where the real norming factor Ν~ is
[ ]
[ ]
+
+
=Ν
−
−
.)(~
)(~
2
)1(~
)1(~
2~
01101
1
oddisnifUBUA
evenisnifUBUA
nn
nn
ωω
(17)
VIII. EXAMPLE
Design the ER HB FIR filter specified by the minimal
passband frequency πω 45.0=Tp and by the minimal
attenuation in the stopband dBas 120−= .
We get 393856.38 →=n (11), 15571103.0=′κ
(12), A = 1.17117396 (13), B = 0.83763199 (14) and Ν~ =
-2747.96038544 (17). The impulse response )(kh
(Tab.2) with the length N=159 coefficients is evaluated using Tab.1 and (10). The actual values
πω 4502.0=Tactp and 91.120−=acta dB satisfy the filter
specification. The amplitude frequency response
)(log20 jwTeH [dB] of the filter is shown in Fig.5. The
detailed view of its passband is shown in Fig.6. For the
specified values πω 45.0=Tp and N=159, the compa-
rative numerical design based on the "Trick" [2]
IWSSIP 2010 - 17th International Conference on Systems, Signals and Image Processing
56
TABLE II. COEFFICIENTS OF THE IMPULSE RESPONSE
Figure 4. )(ωQ for odd and even n
combined with the Remez algorithm using the Matlab function firpm results in the slightly unsatisfactory min-
imal passband frequency ππω 45.044922001.0 <=Tactp
and consequently in a slightly better minimal attenuation
in the stopband 29066608.123−=acta [dB].
Figure 5. Amplitude frequency response )(log20jwT
eH
Figure 6. Passband of the filter
IX. CONCLUSIONS
This paper has presented the equiripple approximation of halfband FIR filters and a novel efficient non-numerical method for their design.
ACKNOWLEDGEMENTS
This work was supported by the grant No.
MSM6840770014, Ministry of Education, Czech
Republic.
REFERENCES
[1] J.H. McClellan, T. W. Parks, L. R. Rabiner, A Computer Program for Designing Optimum FIR Linear Phase Digital Filters, IEEE Trans. Audio Electroacoust, Vol. AU - 21, Dec. 1973, pp. 506 - 526.
[2] P. P. Vaidyanathan, T. Q. Nguyen, A "TRICK" for the Design of FIR Half-Band Filters, IEEE Transactions on Circuits and Systems, Vol. CAS - 34, No. 3, March 1987, pp. 297 - 300.
[3] P. N. Wilson, H. J. Orchard, A Design Method for Half-Band FIR Filters, IEEE Transactions on Circuits and Systems - I: Fundamental Theory and Applications, Vol. CAS - 46, No. 1, January 1999, pp. 95 - 101.
[4] P. Zahradnik, M. Vlcek, R. Unbehauen, Almost Equiripple FIR Half-Band Filters, IEEE Transactions on Circuits and Systems - I: Fundamental Theory and Applications, Vol. CAS - 46, No. 6, June 1999, pp. 744 – 748.